One-loop partition functions in $T\overline{T}$-deformed AdS$_3$

We study the geometry of $T\bar{T}$-deformed BTZ black hole and find it can be regarded as a quotient of hyperbolic space. We then consider the massive scalar field propagating in the $T\bar{T}$-deformed BTZ black hole background. The one-loop partition function of scalar field is calculated using the heat kernel method and the Wilson spool proposal. These two methods give consistent result which implies the Wilson spool proposal still holds under $T\bar{T}$ deformation. Moreover, we also calculate the one-loop partition function of graviton in $T\bar{T}$-deformed BTZ black hole. We find the deformed one-loop partition functions are modified in a simple way, which corresponds to a replacement of the modular parameter. The result precisely matches the large $c$ expansion of $T\bar{T}$-deformed CFT partition function. These results provide a further check about the correspondence between $T\bar{T}$-deformed CFT$_2$ and AdS$_3$ with mixed boundary condition.